We provide a gentle alert that the standard definitions of basic algebraic structures--such as that of a group--contain aspects that may be questionable to students, and discuss what instructors could do to minimise the questionability.

I provide a visual proof for the "Convergence of a Hyper power sequence," which generalises a beautiful result; also proved visually by Azarpanah in 2004.

In this short note I present an elementary proof of irrationality for the number "e," the base of the natural logarithm. It is simpler than other known proofs as it does not use comparisons with geometric series, nor Beukers' integrals, and it does not assume that "e" is a rational number from the beginning.

Anaerobic digestion is a complex biochemical process that is represented by the stoichiometric Buswell's equation. Buswell's equation is the study of the interface among biochemistry, mathematics, and chemistry. Students often have difficulty in understanding the nature of biochemical stoichiometry. Buswell's equation is significant for the…

We view the (real) Laplace transform through the lens of linear algebra as a continuous analogue of the power series by a negative exponential transformation that switches the basis of power functions to the basis of exponential functions. This approach immediately points to how the complex Laplace transform is a generalisation of the Fourier…

Most methods for structural equation modeling (SEM) focused on the analysis of covariance matrices. However, "Historically, interesting psychological theories have been phrased in terms of correlation coefficients." This might be because data in social and behavioral sciences typically do not have predefined metrics. While proper methods…

In XVII century, presumably between 1637 and 1638, with a note in the margin of Diophantus' "Arithmetica", Pierre de Fermat stated that Diophantine equations of the Pythagorean form, x[superscript n] + y[superscript n] = z[superscript n], have no integer solutions for n > 2, and (x, y, z) > 0. Of this statement, however, Fermat…

This case study used individual interviews to investigate graduate students' sensemaking in upper-level electrostatics in the context of problems that can be efficiently solved for the electric potential using Laplace's equation. Although there are many technical mathematical issues involved in solving Laplace's equation, the focus of this…

The random intercept cross-lagged panel model (RICLPM) decomposes longitudinal associations between two processes X and Y into stable between-person associations and temporal within-person changes. In a recent study, Bailey et al. demonstrated through a simulation study that the between-person variance components in the RICLPM can occur only due…

Multilevel factor mixture modeling (FMM) is a hybrid of multilevel confirmatory factor analysis (CFA) and multilevel latent class analysis (LCA). It allows researchers to examine population heterogeneity at the within level, between level, or both levels. This tutorial focuses on explicating the model specification of multilevel FMM that considers…

Mathematics provides us with tools to capture and explain phenomena in everyday biology, even at the nanoscale. The most regularly applied technique to biology is differential equations. In this article, we seek to present how differential equation models of biological phenomena, particularly the flow through ion channels, can be used to motivate…

In this paper, we discuss how a professor could have students, using rational and exponential functions, develop models for several different density-dependent populations based on characteristics of the species. Students would then need to choose an appropriate method from among three different analytic methods (find a solution, phase-line…

Differential equations (DEs) are a powerful tool for modeling real-world contexts. Most research in this area has examined students' understanding and reasoning with pre-packaged DEs, with little attention being given to setting up sophisticated DEs to model complicated real-world situations. This study contributes through a collective case study…

Imaginary latent variables are variables with negative variances and have been used to implement constraints in measurement models. This article aimed to advance this practice and rationalize the imaginary latent variables as a method to detect possible latent deficiencies in measurement models. This rationale is based on the theory of complex…

Students often instrumentally use variables and unknowns without considering the variational thinking behind them. Using parameters to modify the coefficients or unknowns in equations or systems of linear equations (without altering their structure) involves consciously incorporating variational thinking into problem-solving. We will test the…